Bond Graph Modeling and PID Controller ...

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tensions in the muscle fibers and sends proprioceptive feedback to the muscle ... Muscle spindle consists of two intrafusal bags and chain. [5],[6]. Model for bags ...
Bond Graph Modeling and PID Controller Stabilization of Single Link Mechanical Model Madiha Zoheb1 & A. Mahmood2 Department of Electrical and Computer Engineering Center for Advanced Studies in Engineering (CASE) Islamabad, Pakistan [email protected], [email protected]

Abstract— A basic biomechanical model for musculoskeletal structure consists of bones, muscles, muscle spindle and GTO. Muscles are attached to bones by GTO. The bones act as an anchor that muscles can pull on in order to create movement and muscle spindle is responsible for connecting this structure with brain. The dynamic effects of muscle strength, timing of muscle activations, and body geometry have been modeled for a wide variety of human activities. These types of models require the development of complex system equations that account for the effects of rigid-body dynamics, passive and active resistance to motion, and other physiological structures. Bond graph technique is an effective and simple tool for obtaining system equation of any biomechanical system. In this paper, we used bond graph methodology to integrate the single model with 6th order muscle spindle and a GTO, using 20-Sim. The single link nonlinear model sends feedback to CNS through muscle spindle and GTO. We model the closed loop with a PID controller and get results for movement correction. Our simulation results shows that CNS responds to proprioceptive feedback loop for controlling motion. Keywords— musculoskeletal structure, single link model, bond graph, PID controller, prioprioceptive feedback.

I.

INTRODUCTION

One of the major issues in Biomedical engineering field is creating mathematical models that resemble the human body, in a manner that gives us the opportunity to recreate, simulate or analyze movements like walking, running or stepping over obstacles. Physiological systems have been subject of interest to a variety of individuals including artists, physicians, athletes, scientists and researchers. The use of models to simulate human locomotion is well established. Modeling can provide a unique perspective on the underlying control issues, in that the model is neither restricted by experimental procedures nor subject to variability. However, the model should reflect the observed kinematics and dynamics of the system it represents. Bond-graph modeling is a powerful tool for modeling engineering systems, especially when different physical domains are involved. In our work we used the bond graph modeling approach to study the biomechanical system. Primary elements of musculoskeletal structure are muscles, Golgi tendon organs (GTO) and muscle spindle [1].

Muscles and tendons actuate movement by developing and transmitting force to the skeleton. Muscle spindle is responsible for communication with brain. Different bond graphs are available for all these elements which will be discussed briefly in this section. A primary object of interest in the physiological system is the muscle structure; researchers have studied muscles from varying prospective and developed multidimensional understanding of its structure. Early biomechanics researchers understood the muscle as a springlike element and modeled it as stiff object. Later this model was improved in the form of Hill type muscle model (1938, 1950). Hill presented muscle structure as a combination of contractile and a pair of stiff elements in series or parallel arrangement [2]. There are two types of hill’s model. In both models elements are same but the structure is different. Second primary element of musculoskeletal structure is bone. In the simplest scenario bone can be treated as inertia in a mechanical system. If we consider more sophisticated approach, we can compare bone structure in single link model as inverted pendulum mechanical system. This system forms nonlinear bond graph. The inverted pendulum model will be discussed in detail in next section. When muscles contract it develops forces which are applied in series with tendon organs [3]. GTO responds to tensions in the muscle fibers and sends proprioceptive feedback to the muscle for activation. GTO is usually represented by static gains as in [4] but in bond graph detailed models of GTO can be implemented. In a simple approach GTO can be modeled as only elastic spring element or combination of elastic element with viscous damping with negligible mass. Muscle spindles are activated by neural excitation of dynamic and static fusimotors and mechanical inputs of fascicle length and fascicle velocity. The outputs of muscle spindle are primary and secondary afferents which modulate the descending neural excitation commands to the muscle. Muscle spindle consists of two intrafusal bags and chain [5],[6]. Model for bags and chains are available which are connected to form complete bond graph for muscle spindle. In this paper we will be integrating available bond graph models together to form state space of single link model of human leg for sit to stand motion. Then this state space is

transferred to Matlab Simulink and PID controller is designed to stabilize the model response. II.

METHODOLOGY

One way in which model refinement can be expedited is through the use of bond graph modeling techniques [7],[8]. Bond graph elements, which focus on the transfer of energy in terms of effort (force) and flow (velocity), can be combined in order to incorporate all parts of a dynamic system, whether mechanical, electrical, hydraulic, or thermodynamic. The graphical representation of the model quickly leads to the production of overall system equations, numerical solution of these equations, and analysis of the system output. These modeling techniques can also be used to apply a more algorithmic approach to systems modeling, in which the addition or subtraction of separate modules can alter the level of detail in an initial attempt at modeling the system in question. While bond graph techniques have found extensive application in a broad variety of applications, they have been used only sparingly in the field of biomedical engineering. An extensive literature search revealed only a few studies involving the modeling of articular cartilage [9], the cardio vascular system [10], a mobile two-legged robot [11], lung dynamics [12], biological heat and mass transfer [13,14], drug release and simple ergonomic factors of vehicle interiors . Models of generalized musculoskeletal systems have not been published, despite these system’s suitability for use with a modular systems modeling approach. This paper contains integration of bond graphs of basic elements of single link musculoskeletal structure in order to obtain its state space .This task is performed using 20-sim. In the proceeding sections bond graphs of Muscles, Inertia, GTO and muscle spindle are briefly. Then these models are integrated together to form bond graph of complete single link structure. A) MUSCLE MODEL Most appropriate model available for muscles are Hill type models. There are two types of hill type model. Both contain same elements but the arrangement is different. In this paper we will be using 2nd Hill type model. The reason for not using 1st Hill type Model is to avoid differential causality in the bond graph. There are four basic elements in Hill type model. The model is constituted by a contractile element (CE), two non-linear spring elements, one in series (SE) and another in parallel (PE), a damper and two Inertial elements. The description of these elements is as follows: Contractile element (CE): It corresponds to the role played by voltage in electronic circuits [2]. The active force of the contractile element comes from the force generated by the actin and myosin crossbridges at the sarcomere level. 

restoring force that tends to return the muscle to its original length. It is this elastic restoring force that is represented by elastic elements. We may regard the Cp as being mostly due to connective tissues and Cs as being primarily dominated by tendon fibers.

The Elastic elements (parallel and series spring): A muscle when passively stretched exhibits an elastic



The damper element (B):-It is an empirical factor that muscle tension during contraction and speed of contraction are coupled together. Such behavior is represented by a damper.  The Inertial Elements: - Two inductors are used in the Bond graph I1 and I2. I1 is used for mass of Muscle and I2 is used as mass of human body. The bond graph for 2nd Hill type Muscle model is as follows.

Figure 1:- Bond Graph for 2nd Hill type model

B) BONES STRUCTURE Most simple structure for bones is Inertia (I). But if we consider a more sophisticated approach, we can compare single link bone structure with inverted pendulum. Figure 2 shows inverted pendulum system.

Figure 2:- Single link inverted pendulum biomechanical model

A single-link with a triangular base of support, with mass m, inertia I, torque τ and a distance k from center of mass to a joint was discussed in [2]. In this 2nd order nonlinear model, angle θ and angular velocity θ́ are state variables. Output to this system is angular velocity θ́ and angular acceleration θ́ ́ .The nonlinear state space representation of the inertial subsystem is given as

 (t)   (t)    0      I mgk sin  (t)      1   (t)  I mk  

(1)

graph is designed. Bond graphs of muscles, GTO, bones and muscle spindle act as sub models which are integrated together to form larger system.

The bond graph of an inverted pendulum model has been discussed in detail in several textbooks such as [15]. C) GOLGI TENDON ORGAN (GTO) GTO responds to tensions in the muscle fibers and sends proprioceptive feedback to the muscle for activation. GTO is usually represented by static gains as in [4] but in bond graph detailed models of GTO can be implemented. In a simple approach GTO can be modeled as only elastic spring element or combination of elastic element with viscous damping with negligible mass. D) MUSCLE SPINDLE Muscle spindles are activated by neural excitation of dynamic (  d) and static (  s) fusimotors and mechanical inputs of fascicle length and fascicle velocity. The outputs of muscle spindle are primary and secondary afferents which modulate the descending neural excitation commands to the muscle. The model structure for the muscle spindle from [6] is shown in Fig.3 with two intrafusal bags and chain.

Figure 3:- Muscle Spindle Model

Structure of each intrafusal bag remains the same; output of bag 1 is the primary afferent and output of bag 2 and the chain are the secondary afferent. These outputs are functions of extrafusal contractile length, dynamics of intrafusal fiber, mechanical and neural inputs. Each of intrafusal bag and chain consists of a contractile element and series and parallel stiffness (spring) elements. Mechanical inputs are the fascicle length and velocity from musculoskeletal structure. The neural outputs primary (Ia) and secondary afferent (II) from the muscle spindle adds into neural excitation Nin of the muscle.

III.

SINGLE LINK BIOMECHANICAL MODEL

The Block diagram for single link model is given in figure 4. This figure acts as basic model on the basis of which bond

Figure 4:- Model for single link structure

Fig 5 shows the Bond Graph of single link model displayed above. The aim of this paper is to design a model for sit to stand motion. Using standing position as reference (θ=0 degrees) the basic condition we are considering is that initially bone is tilted by 10 degrees (θ=10 degrees) and we have to stabilize the system at standing position. The reference input in Fig.5 is taken as zero. The reason for such reference input is that our final position is static. Input signal is fed to 2nd Hill type Muscle model. The values of B, Cs, Cp, I1 and I2 are taken from [11]. In this muscle model I1 is the weight of muscle and I2 is mass of human body. There is a 10% tolerance margin in the mass of the body. This tolerance is catered as disturbance in the system. This disturbance is presented in the system as modulated source of effort “constant 1”. GTO senses changes in muscle tension and gives feedback to brain. In our case brain will be PID controller which will be discussed in the next section. GTO model is given in [11]. Input to this model is flow from muscle structure. The output to this model is message to brain. Output from muscle model is translational flow and effort. This muscle model is connected to bone structure which requires rotational input. So a transformer is used to convert translational force from muscle structure to torque. This output along with constant1 source of effort is fed to bone structure. Bone structure is similar to inverted pendulum system. The equation of inverted pendulum system is well established [2]. This model is nonlinear in nature. The values of all the elements used in the model are given in table II. Bone structure in second order system with angle θ and angular velocity θ’ are two state variables. When these state variables are connected to transformers we get translation parameters i.e. fascicle length and fascicle velocity. These two parameters are input to muscle spindle structure. Fig. 7 presents complete muscle spindle model with two bags and one ring structure. The outputs of muscle spindle are primary and secondary afferents this basically gives information to the brain about muscle length (stretched length).The Bond graph for muscle spindle model is give in [11].-

Figure 5:- Bond graph of Single Link Model

There are total three outputs, one from GTO and two from muscle spindle to be sent to the brain so all these three outputs are combined in a zero junction that is a common flow junction. And the output is flow of resistor that is connected to zero junction.

spindle w3. These three components form the column matrix w. Reference input to the system is zero as the stable position of system is stationary. The output of system is flow of Rout which is actually sum of all the messages that are to be sent to brain. The Bode plot of the system is given below.

IV. RESULTS AND SIMULATIONS The bond graph gives us state space matrices of the system. The output of bond graph designed in 20 sim gives Bode plot and state space. The state space equation of single link system is

x  t   A x  t   B u  t   Bw w  t  y t   C x t   D u t 

(2) (3)

In the above equations A, B, C, D are the state matrices. Bʷ is the disturbance matrix. There are three disturbances catered in this system. First is tolerance in weight w1, second is dynamic input to muscle spindle w2 and third is static input to muscle

Figure 8:- Bode Plot of Single Link Model

The plot in figure 8 shows that system is unstable. A controller is required to stabilize the system. There are two assumptions for controller design. Disturbances will not be considered and as there are four pole zero cancellation so we will consider minimum realization for controller design. The stable output after designing and implementing PID controller is given as follows.

Figure 9:- Stable System Output after Using PID Controller

The values of controller variables are given in Table I. PID controller acts as brain in the system. A more sophisticated controller can be designed by considering the disturbances. V. CONCLUSION Computational models of biomechanical systems are critical tools in reaching new understanding of the workings of the human body. Despite the fact that these types of models have been extremely useful in any number of biomedical fields, there has been little effort to standardize the modeling tools used by investigators studying biomechanical systems. The incorporation of bond graph techniques into biomechanical modeling would aid in streamlining and standardizing the modeling process. We have analyzed our single link bio mechanical model with bond graph and have stabilized with PID controller which acts as a CNS. In this paper we developed a complete feedback model using a nonlinear single model for motion stabilization. Muscle sends feedback to CNS through GTO and muscle spindle through 6th order muscle spindle. These muscle spindle output collected at afferents and GTO output passed to PID controller as CNS. In actual system the loop has multiple loops in CNS for better motion coordination. In future we will extend our work to three link biomechanical model with proprioceptive feedback at each joint to control movement coordination tasks. APPENDIX TABLE I. Specifications of PID controller S.No

Parameters

Values

1

P

0

2

I

2256.9

3

D

0

4

N

100

TABLE II. Definition of Single Link Model Terms Elements names B, B bag1, Bbag2 2, B ring B gto Constant Constant 1 Constant 2 Cp Cs Gain 1 I1 I2 Ia, II K bag1, bag 2, gto K ring Ksr1, Ksr2, Ksr3 m_TF_i TF, TF1 Cosine Cosine 1

Values 1200 Ω 3200 Ω 89.4 0.0 0.1 0.0003125 F 0.0000101 F 768.32 1H 70 H 1200 H 0.001 F 0.001 F 0.00025 F 0.625 1 Amp=1 Amp=0.1

Omega=1 rad/s Omega=546.64 rad/s

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